3 Circles

Let $r , R_1 , R_2$ be the radius of the inner circle, middle and outer circle respectively.

Then according to the question:

• $\large \displaystyle \pi (R_1^2 - r^2) = \pi r^2\\ \large \displaystyle R_1^2 - 3 = 3 \implies R_1 = \sqrt6.$

• $\large \displaystyle \pi (R_2^2 - R_1^2) = \pi r^2\\ \large \displaystyle R_2^2 - 6 = 3 \implies R_2 = 3.$

$\large \displaystyle \therefore \text{The radius of the largest circle is } \color{#D61F06}{\boxed{3}}.$

Let ${ A }_{ 1 }$ , ${ A }_{ 2 }$ , ${ A }_{ 3 }$ be the area of the smaller circle, the one in the middle and the larger, respectively. We can write the given information as:

• ${ A }_{ 1 }={ A }_{ 3 }-{ A }_{ 2 }$

• ${ A }_{ 1 }={ A }_{ 2 }-{ A }_{ 1 }$

Thus, $2{ A }_{ 1 }={ A }_{ 2 }$ . Therefore, ${ A }_{ 3 }=3{ A }_{ 1 }=3\cdot \pi \cdot { \sqrt { 3 } }^{ 2 }=9\pi$ . Hence, the radius of the larger circle is $3$ .